In this report we treat nonlinear programs Pro(f, h;K) having an objective function f , a finite number of equality constraints h(x) = (h1(x), · · · , hl(x)) = 0, and an abstract convex constraint x ∈ K with its convex set K. Our particular interest is an algebraic criterion for a locally isolated stationary solution to be strong stable, in the sense of Kojima, under a Linear Independence Const...

Given a finite set F of estimators, the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in F . It was conjectured that empirical minimization performed in the convex hull of F is an optimal aggregation method, but we show that this conjecture is false. Despite that, we prove that empirical minimization in the convex hul...

In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly α(·)-paraconvex. Let (X, ‖.‖) be a real Banach space. Let f(x) be a real valued strongly α(·)-paraconvex function defined on an open convex subset Ω ⊂ X , i.e. f ( tx+(...

Abstract Given a finite set F of estimators, the problem of aggregation is to construct a new estimator that has a risk as close as possible to the risk of the best estimator in F . It was conjectured that empirical minimization performed in the convex hull of F is an optimal aggregation method, but we show that this conjecture is false. Despite that, we prove that empirical minimization in the...

We consider optimizing a function smooth convex function f that is the average of a set of differentiable functions fi, under the assumption considered by Solodov [1998] and Tseng [1998] that the norm of each gradient f ′ i is bounded by a linear function of the norm of the average gradient f . We show that under these assumptions the basic stochastic gradient method with a sufficiently-small c...

In this paper we deal with the following generalized vector variational inequality problem: let X, Y and Z be topological vector spaces, K be a convex set in X, D be a nonempty set in Y , and C be a closed convex cone in Z. Let T : K → 2 be a multifunction and f : K ×D ×K → Z be a single-valued mapping. Find a point x̂ ∈ K and ŷ ∈ T (x̂) such that f(x̂, ŷ, z) / ∈ −intC, ∀z ∈ K. We prove some exist...

We are given data α1, . . . , αm and a set of points E = {x1, . . . , xm}. In this paper we address the question of conditions ensuring the existence of a function f satisfying the interpolation conditions f(xi) = αi, i = 1, . . . ,m, that is also n-convex on a set properly containing E. We consider both one point extensions of E, and extensions of E to all of IR. We also determine bounds on n-...

In this paper, we present a new iterative scheme for finding a common element of the solution set F of the split feasibility problem and the fixed point set F(T) of a right Bregman strongly quasi-nonexpansive mapping T in p-uniformly convex Banach spaces which are also uniformly smooth. We prove strong convergence theorem of the sequences generated by our scheme under some appropriate condition...

In a convex optimization problem, x ∈ R is a vector known as the optimization variable, f : R → R is a convex function that we want to minimize, and C ⊆ R is a convex set describing the set of feasible solutions. From a computational perspective, convex optimization problems are interesting in the sense that any locally optimal solution will always be guaranteed to be globally optimal. Over the...

In the paper a class of families F(M) of functions defined on differentiable manifolds M with the following properties: 1F . if M is a linear manifold, then F(M) contains convex functions, 2F . F(·) is invariant under diffeomorphisms, 3F . each f ∈ F(M) is differentiable on a dense Gδ-set, is investigated.